Yes. Online banking is absolutely safe to do online and is done by millions daily. Below are some of the different ways to keep online banking safe or ways it may become compromised.
Many ways attackers get into someone's bank account is by tricking a user into thinking they're logging into their account when their really not. This technique is known as phishing and is often done through e-mail, where it informs you that you need to change your online information, verify a purchase, or something else that would want to make you log into your online account through a link in the e-mail. In reality the link points to a fake page that logs your account information.
This attack can be avoided by never clicking on a link to visit your online bank. Instead if you need to log into your bank always visit it through a bookmark you created in your browser or by typing in the URL.
See the phishing definition for real life examples and other suggestions in preventing phishing attacks.
Make sure the page is secure when entering data
Any time you visit a web page that is encrypting information a small lock () will appear either in the bottom corner of the window or next to the address bar and the URL will start with https:// instead of http://. If you do not see this all data is not secure and anything you enter into the page could be captured and read by someone.
When you visit your online bank login page verify you see this lock before entering your username and password. If you do not see this, do not log into the page.
Never send usernames, passwords, etc. through e-mail
No bank will ever request for you to send personal information over e-mail. Never send your username, password, PIN, account information, credit card, etc. over e-mail. E-mail is unencrypted and if intercepted by a third-party could be read. It's also often stored on a server; if that server was to become compromised the attacker could read that e-mail with your personal information.
Be cautious where you log into your bank
For most users we suggest only logging into your online bank page while at home. Your place of work can install key loggers or use other methods of monitoring you while online. Someone who has access to this information could access these logs that can contain all keystrokes including usernames and passwords.
When on a wireless network it's important to realize that all information being sent to and from your computer to the wireless router can be intercepted and read by someone nearby. Therefore if you need to log into your online bank page while on a wireless network make sure the network your connecting to is secure using WPA.
Be concerned when logging into an account from a friends computer. A computer you're not familiar with could intentionally or unintentionally log usernames and passwords that could be used to access your account.
Use a strong password
Your password you're using to log into your online bank should be a strong password. What this means is that it shouldn't be something easily guessed such as your pets name. Your banking password should have number, special character, and different capitalization. See the password definition for additional information, links, and examples.
Make sure your computer is protected
Finally, it's always a good idea to keep your personal computer protected, when an attacker attacks or infects a computer they could install a key logger that logs each keystroke you enter into a computer. These can capture your username, password, and other confidential data.